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2-D quadratic maps and 3-D ODE systems [electronic resource] : a rigorous approach / Elhadj Zeraoulia, Julien Clinton Sprott.

By: Zeraoulia, Elhadj.
Contributor(s): Sprott, Julien C.
Material type: TextTextSeries: World Scientific series on nonlinear scienceSeries AMonographs and treatises: v. 73.Publisher: Singapore ; Hackensack, N.J. : World Scientific, c2010Description: 1 online resource (xiii, 342 p.) : ill.ISBN: 9789814307758 (electronic bk.); 9814307750 (electronic bk.).Subject(s): Forms, Quadratic | Differential equations, Linear | Bifurcation theory | Differentiable dynamical systems | Proof theory | Mathematics | MATHEMATICS -- Differential Equations -- OrdinaryGenre/Form: Electronic books. | Electronic books.Additional physical formats: Print version:: 2-D quadratic maps and 3-D ODE systems.DDC classification: 515.352 Online resources: EBSCOhost
Contents:
1. Tools for the rigorous proof of chaos and bifurcations. 1.1. Introduction. 1.2. A chain of rigorous proof of chaos. 1.3. Poincare map technique. 1.4. The method of fixed point index. 1.5. Smale's horseshoe map. 1.6. The Sil'nikov criterion for the existence of chaos. 1.7. The Marotto theorem. 1.8. The verified optimization technique. 1.9. Shadowing lemma. 1.10. Method based on the second-derivative test and bounds for Lyapunov exponents. 1.11. The Wiener and Hammerstein cascade models. 1.12. Methods based on time series analysis. 1.13. A new chaos detector. 1.14. Exercises -- 2. 2-D quadratic maps : The invertible case. 2.1. Introduction. 2.2. Equivalences in the general 2-D quadratic maps. 2.3. Invertibility of the map. 2.4. The Henon map. 2.5. Methods for locating chaotic regions in the Henon map. 2.6. Bifurcation analysis. 2.7. Exercises -- 3. Classification of chaotic orbits of the general 2-D quadratic map. 3.1. Analytical prediction of system orbits. 3.2. A zone of possible chaotic orbits. 3.3. Boundary between different attractors. 3.4. Finding chaotic and nonchaotic attractors. 3.5. Finding hyperchaotic attractors. 3.6. Some criteria for finding chaotic orbits. 3.7. 2-D quadratic maps with one nonlinearity. 3.8. 2-D quadratic maps with two nonlinearities. 3.9. 2-D quadratic maps with three nonlinearities. 3.10. 2-D quadratic maps with four nonlinearities. 3.11. 2-D quadratic maps with five nonlinearities. 3.12. 2-D quadratic maps with six nonlinearities. 3.13. Numerical analysis -- 4. Rigorous proof of chaos in the double-scroll system. 4.1. Introduction. 4.2. Piecewise linear geometry and its real Jordan form. 4.3. The dynamics of an orbit in the double-scroll. 4.4. Poincare map [symbol]. 4.5. Method 1 : Sil'nikov criteria. 4.6. Subfamilies of the double-scroll family. 4.7. The geometric model. 4.8. Method 2 : The computer-assisted proof. 4.9. Exercises -- 5. Rigorous analysis of bifurcation phenomena. 5.1. Introduction. 5.2. Asymptotic stability of equilibria. 5.3. Types of chaotic attractors in the double-scroll. 5.4. Method 1 : Rigorous mathematical analysis. 5.5. Method 2 : One-dimensional Poincare map. 5.6. Exercises.
Summary: This book is based on research on the rigorous proof of chaos and bifurcations in 2-D quadratic maps, especially the invertible case such as the H�non map, and in 3-D ODE's, especially piecewise linear systems such as the Chua's circuit. In addition, the book covers some recent works in the field of general 2-D quadratic maps, especially their classification into equivalence classes, and finding regions for chaos, hyperchaos, and non-chaos in the space of bifurcation parameters. Following the main introduction to the rigorous tools used to prove chaos and bifurcations in the two representative systems, is the study of the invertible case of the 2-D quadratic map, where previous works are oriented toward H�non mapping. 2-D quadratic maps are then classified into 30 maps with well-known formulas. Two proofs on the regions for chaos, hyperchaos, and non-chaos in the space of the bifurcation parameters are presented using a technique based on the second-derivative test and bounds for Lyapunov exponents. Also included is the proof of chaos in the piecewise linear Chua's system using two methods, the first of which is based on the construction of Poincare map, and the second is based on a computer-assisted proof. Finally, a rigorous analysis is provided on the bifurcational phenomena in the piecewise linear Chua's system using both an analytical 2-D mapping and a 1-D approximated Poincare mapping in addition to other analytical methods.
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Includes bibliographical references and index.

1. Tools for the rigorous proof of chaos and bifurcations. 1.1. Introduction. 1.2. A chain of rigorous proof of chaos. 1.3. Poincare map technique. 1.4. The method of fixed point index. 1.5. Smale's horseshoe map. 1.6. The Sil'nikov criterion for the existence of chaos. 1.7. The Marotto theorem. 1.8. The verified optimization technique. 1.9. Shadowing lemma. 1.10. Method based on the second-derivative test and bounds for Lyapunov exponents. 1.11. The Wiener and Hammerstein cascade models. 1.12. Methods based on time series analysis. 1.13. A new chaos detector. 1.14. Exercises -- 2. 2-D quadratic maps : The invertible case. 2.1. Introduction. 2.2. Equivalences in the general 2-D quadratic maps. 2.3. Invertibility of the map. 2.4. The Henon map. 2.5. Methods for locating chaotic regions in the Henon map. 2.6. Bifurcation analysis. 2.7. Exercises -- 3. Classification of chaotic orbits of the general 2-D quadratic map. 3.1. Analytical prediction of system orbits. 3.2. A zone of possible chaotic orbits. 3.3. Boundary between different attractors. 3.4. Finding chaotic and nonchaotic attractors. 3.5. Finding hyperchaotic attractors. 3.6. Some criteria for finding chaotic orbits. 3.7. 2-D quadratic maps with one nonlinearity. 3.8. 2-D quadratic maps with two nonlinearities. 3.9. 2-D quadratic maps with three nonlinearities. 3.10. 2-D quadratic maps with four nonlinearities. 3.11. 2-D quadratic maps with five nonlinearities. 3.12. 2-D quadratic maps with six nonlinearities. 3.13. Numerical analysis -- 4. Rigorous proof of chaos in the double-scroll system. 4.1. Introduction. 4.2. Piecewise linear geometry and its real Jordan form. 4.3. The dynamics of an orbit in the double-scroll. 4.4. Poincare map [symbol]. 4.5. Method 1 : Sil'nikov criteria. 4.6. Subfamilies of the double-scroll family. 4.7. The geometric model. 4.8. Method 2 : The computer-assisted proof. 4.9. Exercises -- 5. Rigorous analysis of bifurcation phenomena. 5.1. Introduction. 5.2. Asymptotic stability of equilibria. 5.3. Types of chaotic attractors in the double-scroll. 5.4. Method 1 : Rigorous mathematical analysis. 5.5. Method 2 : One-dimensional Poincare map. 5.6. Exercises.

This book is based on research on the rigorous proof of chaos and bifurcations in 2-D quadratic maps, especially the invertible case such as the H�non map, and in 3-D ODE's, especially piecewise linear systems such as the Chua's circuit. In addition, the book covers some recent works in the field of general 2-D quadratic maps, especially their classification into equivalence classes, and finding regions for chaos, hyperchaos, and non-chaos in the space of bifurcation parameters. Following the main introduction to the rigorous tools used to prove chaos and bifurcations in the two representative systems, is the study of the invertible case of the 2-D quadratic map, where previous works are oriented toward H�non mapping. 2-D quadratic maps are then classified into 30 maps with well-known formulas. Two proofs on the regions for chaos, hyperchaos, and non-chaos in the space of the bifurcation parameters are presented using a technique based on the second-derivative test and bounds for Lyapunov exponents. Also included is the proof of chaos in the piecewise linear Chua's system using two methods, the first of which is based on the construction of Poincare map, and the second is based on a computer-assisted proof. Finally, a rigorous analysis is provided on the bifurcational phenomena in the piecewise linear Chua's system using both an analytical 2-D mapping and a 1-D approximated Poincare mapping in addition to other analytical methods.

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2-D quadratic maps and 3-D ODE systems by Zeraoulia, Elhadj. ©2010
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